# Partial regularity for doubly nonlinear parabolic systems of the first type

## TL;DR

This paper establishes partial regularity results for weak solutions of a class of doubly nonlinear parabolic PDE systems, showing that the spatial gradient is locally Hölder continuous except on a lower-dimensional set.

## Contribution

It introduces a novel partial regularity result for solutions of doubly nonlinear parabolic systems, extending understanding of their regularity properties.

## Key findings

- D{f v} is locally Hölder continuous outside a lower-dimensional set.
- The proof uses compactness, integral identities, and fractional time derivative estimates.
- Results apply to models of phase transitions in materials.

## Abstract

We study solutions ${\bf v}$ of the parabolic system of PDE $$ \partial_t\left(D\psi({\bf v})\right)=\text{div}DF(D{\bf v}). $$ Here $\psi$ and $F$ are convex functions, and this is a model equation for more general doubly nonlinear evolutions that arise in the study of phase transitions in materials. We show that if ${\bf v}$ is a weak solution, then $D{\bf v}$ is locally H\"older continuous except for possibly on a lower dimensional subset of the domain of ${\bf v}$. Our proof is based on compactness properties of solutions, two integral identities and a fractional time derivative estimate for $D{\bf v}$.